This proposal aims at making improvements in the theory of spherical capsid structure commensurate with the recent explosion of low resolution structures as determined by cryo-electron microscopy. We now have no theoretical understanding of the assembly of closed shells and polymorphs, and since the thirty year old Quasi-Equivalence theory has met fatal counter-examples, we do not even understand why icosahedral symmetry is observed, instead of octahedral symmetry or random packing. The numerical "Uniform Spacing" model was developed as a partial explanation of observed capsid patterns, and it successfully reproduces the surface morphologies of the capsids to which it has been applied. The proposed work consists of: 1) applying Uniform Spacing to additional empirical capsid patterns; 2) developing a comprehensive library of theoretical capsid morphologies; 3) simulating capsomeres self- organizing into icosahedral or other symmetry; 4) studying the stability of spherical and polyhedral shells and simulating the formation of closed capsid shells and polymorphic structures. The working hypothesis is that capsids self-assemble via long to intermediate-ranged, non-specific forces which mediate the inter-capsomere interactions and are tuned or switched by solvent conditions or specific external mechanisms. Low resolution capsids are modeled as 'generic' structures, chosen by specific mechanisms from a range of alternatives determined by non- specific interactions; the polymorphism observed for some capsid proteins suggests this viewpoint. The simulations will explore the roles of nucleating structures, solvent conditions, and the form of the inter- capsomere forces. The goal is to discover the principles guiding capsid assemble and morphology, and thus to further the determination and understanding of empirical capsid structures. A new method for representing the capsid has been developed and will be implemented. Both the two-dimensional Uniform Spacing model and the proposed three-dimensional shell stability and assemble model describe capsomeres and their interaction via Fourier series. Such representation allows easy reconfiguration of the capsomere shape and the inter- capsomere force laws, and easy calculation of the net capsomere interaction energy, which determines the fitness of a model capsid. Fourier representation is far more computationally tractable than the usual sort of representation in real space. It affords a natural way to describe non-specific forces, filtering out short-ranged details but retaining the global, form-generating interactions which govern self- assembly. The compact representation allows practical computation of extensive simulations of Euler-Langevin dynamics for systems of model capsomeres.